Determine whether each equation defines y as a function of x. x = (1/3)(y²)

Verified step by step guidance

1. Recall the definition of a function: for each input value of \( x \), there must be exactly one output value of \( y \).
2. Given the equation \( x = \frac{1}{3} y^2 \), try to express \( y \) explicitly in terms of \( x \) by isolating \( y \).
3. Multiply both sides by 3 to get \( 3x = y^2 \).
4. Take the square root of both sides to solve for \( y \), remembering to include both the positive and negative roots: \( y = \pm \sqrt{3x} \).
5. Since for a single \( x \) value there are two possible \( y \) values (positive and negative), the equation does not define \( y \) as a function of \( x \).

Key Concepts

Definition of a Function

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). To determine if an equation defines y as a function of x, we check if for every x there is only one y.

Implicit vs. Explicit Functions

An explicit function expresses y directly in terms of x (y = f(x)), while an implicit function relates x and y without isolating y. Understanding this helps analyze equations like x = (1/3)(y^2) to see if y can be uniquely determined from x.

Vertical Line Test and Solving for y

The vertical line test checks if a graph represents y as a function of x by ensuring vertical lines intersect the graph at most once. Alternatively, solving the equation for y and checking if multiple y-values correspond to a single x-value helps determine if y is a function of x.